Master thesis on Rational Homotopy Theory
https://github.com/Jaxan/Rational-Homotopy-Theory
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33 lines
1.7 KiB
33 lines
1.7 KiB
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\subsection{$A$ and $K$ form a Quillen pair}
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We will prove that $A$ preserves cofibrations and trivial cofibrations. We only have to check this fact for the generating (trivial) cofibrations in $\sSet$. Note that the contravariance of $A$ means that a (trivial) cofibrations should be sent to a (trivial) fibration.
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\begin{lemma}
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$A(i) : A(\Delta[n]) \to A(\del \Delta[n])$ is surjective.
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\end{lemma}
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\begin{proof}
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Let $\phi \in A(\del \Delta[n])$ be an element of degree $k$, hence it is a map $\del \Delta[n] \to \Apl^k$. We want to extend this to the whole simplex. By the fact that $\Apl^k$ is Kan and contractible we can find a lift $\overline{\phi}$ in the following diagram showing the surjectivity.
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\cimage[scale=0.5]{Extend_Boundary_Form}
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\end{proof}
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\begin{lemma}
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$A(j) : A(\Delta[n]) \to A(\Lambda^n_k)$ is surjective and a quasi isomorphism.
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\end{lemma}
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\begin{proof}
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As above we get surjectivity from the Kan condition. To prove that $A(j)$ is a quasi isomorphism we pass to the singular cochain complex and use that $C^\ast(j) : C^\ast(\Delta[n]) \we C^\ast(\Lambda^n_k)$ is a quasi isomorphism. Consider the following diagram and conclude that $A(j)$ is surjective and a quasi isomorphism.
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\cimage[scale=0.5]{A_Preserves_WCof}
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\end{proof}
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Since $A$ is a left adjoint, it preserves all colimits and by functoriality it preserves retracts. From this we can conclude the following corollary.
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\begin{corollary}
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$A$ preserves all cofibrations and all trivial cofibrations and hence is a left Quillen functor.
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\end{corollary}
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\begin{corollary}
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$A$ and $K$ induce an adjunction on the homotopy categories:
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$$ \Ho{\sSet} \leftadj \opCat{\Ho{\CDGA}}. $$
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\end{corollary}
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